X-ray characterization by energy-resolved powder diffraction

A method for single-shot, nondestructive characterization of broadband x-ray beams, based on energy-resolved powder diffraction, is described. Monte-Carlo simulations are used to simulate data for x-ray beams in the keV range with parameters similar to those generated by betatron oscillations in a laser-driven plasma accelerator. The retrieved x-ray spectra are found to be in excellent agreement with those of the input beams for realistic numbers of incident photons. It is demonstrated that the angular divergence of the x rays can be deduced from the deviation of the detected photons from the Debye-Scherrer rings which would be produced by a parallel beam. It is shown that the angular divergence can be measured as a function of the photon energy, yielding the angularly resolved spectrum of the input x-ray beam.

Figure 1

Geometry of an XCERP diffraction measurement. A broadband x-ray beam is incident normally on a powdered target. Diffraction from the powder sample produces Debye-Scherrer rings with angular radii which vary with photon energy and lattice plane spacing. The distance along the axis from source to powder is $D_{\mathrm{sp}}$ and the distance from powder to detector is $D_{\mathrm{pd}}$. The radial distances of the detector’s upper and lower edges from the beam axis are $r_1$ and $r_2$, respectively. Most of the x-ray beam is not diffracted and can be used for application such as, in this case, phase-contrast imaging.

Figure 2

Scatter plot of the location $(x,y)$ and energy $E$ of photons arriving in the plane of the detector for the case of diffraction by powdered silicon of an x-ray beam with a synchrotron spectrum of $E_{\mathrm{c}}=20\mathrm{keV}$ and ${\sigma }_{\phi }=2.5\mathrm{mrad}$. The number of photons used in the simulation was $N_{\mathrm{sim}}=1\times 10^5$.

Figure 3

Comparison of the incident and retrieved (blue) spectra for the case of diffraction by powdered silicon; for the incident beam, the underlying spectrum is shown by the dashed black line and that of the sample generated by the Monte-Carlo code is shown by the gray bins. (a) and (b) show the spectra for an incident beam with a synchrotron spectrum with $E_{\mathrm{c}}=20\mathrm{keV}$ and ${\sigma }_{\phi }=2.5\mathrm{mrad}$ and: (a) $1.2\times 10^8\mathrm{photons}$; (b) $6.0\times {10}^6\mathrm{photons}$. Shown in (c) are the incident and deduced spectra for a beam of $1.2\times 10^8\text{photons}$ with a complex spectrum. For all plots the spectrum of the beam transmitted through the powder sample is shown in green, and the error bars (red) are calculated assuming the number of photons detected within each energy bin follows a Poisson distribution.

Figure 4

Scatter plots of radial distance $r$ from the beam axis as a function of photon energy. The solid lines show the radii of the Debye-Scherrer rings $r_{hkl}$ for photons incident along the beam axis. The assumed properties of the incident beam used for Fig. 4 are as in the data shown in Fig. 3, with ${\sigma }_{\phi }=2.5\mathrm{mrad}$. Figures 4 and 4 show the plots for simulations with ${\sigma }_{\phi }=1.0\mathrm{mrad}$ and ${\sigma }_{\phi }=5.0\mathrm{mrad}$, respectively.

Figure 5

Distributions of photons $\mathrm{\Psi }(r)$ for diffraction by powdered silicon and an incident x-ray beam with the same parameters as in Fig. 3 calculated: by the Monte-Carlo diffraction code (grey bars); and for a Gaussian angular distribution ${\mathrm{\Phi }}_{\text{model}}(\phi ,{\sigma }_{\mathrm{test}})$ after least squares minimization (blue). The best fit corresponds to a divergence of ${\sigma }_{\mathrm{test}}=2.51\pm 0.08\mathrm{mrad}$. The red lines show error bars calculated assuming that the number of photons in each radial bin follows a Poisson distribution.

Figure 6

Demonstration of the retrieval of an energy-dependent divergence with XCERP diffraction for an incident beam with the same parameters as Fig. 3 except for an energy-dependent divergence for photons in the range $0\mathrm{keV}<E<50\mathrm{keV}$ given by ${\sigma }_{\phi }[\mathrm{mrad}]=5-0.075E[\mathrm{keV}]$. The retrieved divergence for each 10 keV energy bin is shown in blue; error bars, calculated assuming the number of photons detected within each energy bin are Poisson distributed, are shown by the solid red lines; the standard deviation used to generate the simulation’s incident beam is shown by the dotted green line.

Figure 7

Comparison of the angularly resolved spectrum retrieved by XCERP diffraction (colored surface) with that of an input beam with the parameters as in Fig. 6 (gray mesh). The side panels show the angularly resolved spectra integrated over energy (left) and angle (right) for the retrieved (blue line) and input (gray line) spectrum. The integrated spectra have been scaled so that they can be displayed on the same $z$-axis.